Horizontal lines have zero slope. To graph a horizontal line in the standard coordinate system, use the equation \( y = k ,\) where \(k\) gives the point on the \(y\)-axis that the line will intersect.
Suppose we want a horizontal line that passes through \( (0,5) .\)
In addition to that point, the line will pass through
\[ \ldots, (-3, 5), (-2, 5), (-1, 5), (1, 5), (2, 5), (3, 5), \ldots \]
and in general for any real number \(Q,\) the graph will pass through \( (Q, 5) .\)
This means the \(x\)-coordinate can vary to any real number, so it doesn't get fixed at all and doesn't need to appear in the horizontal line equation. \(y\) on the other hand must always be 5, giving an equation of \( y = 5 .\)
Note that if we attempt to use a traditional line form, like the slope-intercept form \( y = mx + b ,\) we create a line with the equation \(y=0x+b,\) or \(y=b,\) where \(b\) is the \(y\)-intercept of the line.
What is the equation of the line that passes through the points \((5, 4)\) and \((-2,4)\,?\)
Because both of the points on the line have a \(y\)-value of 4, every point on the line will have a \(y\)-value of 4 and the equation of the horizontal line is \(y=4.\)